A very efficient numerical simulation method of the railway vehicle-track dynamic interaction is described. When a vehicle runs at high speed on the railway track, contact forces between a wheel and a rail vary dynamically due to the profile irregularities existing on the surface of the rail. A large variation of contact forces causes undesired deteriorations of a track and its substructures. Therefore these dynamic contact forces are of main concern of the railway engineers. However it is very difficult to measure such dynamic contact forces directly. So it is important to develop an appropriate numerical simulation model and identify structural factors having a large influence on the variation of contact forces.When a contact force is expressed by the linearized Hertzian contact spring model, the equation of motions of the system is expressed as a second-order linear time-variant differential equation which has a time-dependent stiffness coefficient. Applying a well-known Newmark direct integration method, a numerical simulation is reduced to solving iteratively a time-variant, large-scale sparse, symmetric positive-definite linear system.In this study, by defining a special vector named a contact point one, it is shown that this time-variant stiffness coefficient can be expressed simply as a product of the contact point vector and its transpose and so the Sherman-Morrison-Woodbury formula applied for updating the inverse of the coefficient matrix. As a result, the execution of numerical simulation can be carried out very efficiently. A comparison of the computational time is given. Description of railway vehicle-track dynamic interaction by contact point vectorA railway vehicle-track system is excited by contact forces generated beween a wheel and a rail. Fig.1 shows the mathematical model developed to study a railway vehicle-track dynamic interaction. A vehicle model is represented as a single car train with two bogies and four wheels. It runs on the track from left-side to right-side at high speed. In a track model the rail is discretely supported by the sleepers laid on the multi-layer ballasts. Elastic railpads are placed between the rail and the sleepers to attenuate dynamic loads. The rail is represented as an Euler beam with uniform flexural rigidity. The equation of motion of a beam is given by the 4th-order partial differential equation.In this study, according to the linearized Hertzian contact theory, a dynamic contact force , P dyn (t) is represented as being proportional to a total dynamic elastic deformation , ∆ dyn (t) of the rail and the wheel generating at the contact point aswhere k H is a coefficient of linearized Hertzian spring stiffness.As shown in Fig.2, the vehicle has four contact points. To find the dynamic contact force developed at each contact point, the elastic deformation is expressed as 2, 3, 4 (2) y w,i (t) : vertical displacement of the i -th running wheel u w,i (t) : vertical displacement of the rail, just beneath the i -th running wheel r w,i (t) : rail profile irregula...
It is important when studying vehicle-track dynamic interactions to make clear the characteristics of many physical and/or structural parameters of a track. However, it is difficult to estimate them with accuracy only through measurements under a real train load on site. In this paper, we describe an approach using two tools of a test facility and computer simulation. The Track and Structural Dynamics Simulator (TRADYS) is a unique mechanical vibration exciter equipped with hydraulic actuators between a train and a track. The train has the same masses and suspensions as the latest Japanese high speed railway vehicle. We can control arbitrarily the vertical relative displacements between rails and wheels through the directions inputted to the actuator. These relative displacements are supposed to represent a railway track irregularity. In the experiments, dynamic wheel loads, displacements and accelerations of each part of a track are measured as the outputs. In addition to the experiments, computer simulation is carried out. A theoretical model for this special test facility including the actuator is constructed. In this paper the measured physical values and computed solutions are compared, and the accuracy of the parameter values of the track structure components is discussed.
The amplitude of track geometry waveforms is normally used as an index to estimate the geometrical quality of tracks because it is known that track geometry has a correlation with vehicle dynamic responses, such as acceleration, dynamic wheel load and lateral force. To ensure higher quality track maintenance work, however, it is desirable that the index have a higher correlation with vehicle responses. Therefore, we suggest that the dynamic behavior of the vehicle, predicted by means of measured track geometry, can be useful as an index.There are several ways to predict a dynamic system. The theoretical method that involves solving kinematic equations is the most popular 1) , but spectral analysis is also used. However, putting these methods into practice for track maintenance work is beset with problems because of the inherent difficulties in deciding on an accurate mathematical model to use and, for example, in determining the vehicle's stiffness and suspension damper rates. Because these values change over time, the solution to a kinematic equation tends to be different from the measured value when these parameters are inaccurate, and the cost of calculation is high. Another factor is that high-capacity computers are needed for theoretical methods, which is not realistic for practical work carried out at track maintenance depots.Thus, we applied system identification theory to identify the dynamic characteristics of the vehicle to predict how track geometry affected its vertical vibration and dynamic wheel load. The system identification is one of the stochastic signal processing theories similar to spectral analysis, and we can identify the vehicle dynamic characteristics with fewer observed signals than when we use spectral analysis. The main feature of these models is the simplicity of their predictive calculation, which makes it easy to handle these models at track maintenance depots.The predicted vehicle behavior has higher correlation with actual responses and will provide a more suitable track geometry index than amplitude.2. System identification theory 2. System identification theory 2. System identification theory 2. System identification theory 2. System identification theory 2) 2) 2) 2) 2) 2.1 Outline of the system identification 2.1 Outline of the system identification 2.1 Outline of the system identification 2.1 Outline of the system identification 2.1 Outline of the system identification We can approximate most dynamic systems as linear systems. By so doing, we can express the relationship between system input and output signals by using Equation (1).where u(n) is the input signal; y(n) is the output signal; and g(k) is the system impulse response value. The following function G(q) is termed the system's transfer function:where q is the delay operator that has its function expressed by Equation (3).The task of system identification is to estimate a mathematical model of a system based on observed input-output signals. In the case of the estimation of a dynamic vehicle model, the input signal is...
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